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:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''
{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
| Neutral = 2L 6s
}}
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).


'''5L 3s''' refers to the structure of [[MOS]] scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The term '''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') is often used for the octave-equivalent MOS structure 5L 3s, whose brightest mode is LLsLLsLs. The name ''oneirotonic'' (from Greek ''oneiros'' 'dream') was coined by [[Cryptic Ruse]] after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.  
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.


Oneirotonic is a distorted diatonic, because it has one extra small step compared to diatonic ([[5L 2s]]): for example, the Ionian diatonic mode LLsLLLs can be distorted to the Dylathian oneirotonic mode LLsLLsLs.
'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].


Any edo with an interval between 450¢ and 480¢ has an oneirotonic scale. [[13edo]] is the smallest edo with a (non-degenerate) 5L3s oneirotonic scale and thus is the most commonly used oneirotonic tuning.
== Scale properties ==


5L 2s has a pentatonic MOS subset [[3L 2s]] (SLSLL), and in this context we call this the ''oneiro-pentatonic''. When viewed as a chord (with undetermined voicing) we call it the Oneiro Core Pentad. (Note: [[3L 5s]] scales also have 3L 2s subsets.)
=== Intervals ===
{{MOS intervals}}


In terms of [[Tour of Regular Temperaments|regular temperament]]s, there are at least two melodically viable ways to interpret oneirotonic: (see [[5L 3s#Tuning_ranges|Tuning ranges]]):
=== Generator chain ===
# When the generator is between 457.14¢ (8\21) and 461.54¢ (5\13): [[5L_3s#Petrtri_.2813.2621.2C_2.5.9.11.13.17.29|Petrtri]] (13&21, a 2.5.9.11.13.17 temperament that approximates the harmonic series chord 4:5:9:11:13:17)
{{MOS genchain}}
# When the generator is between 461.54¢ (5\13) and 466.67¢ (7\18): [[A-Team]] (13&18, a 2.9.5.21 temperament where two major mosseconds or "whole tones" approximate a [[5/4]] classical major third)
In a sense, these two temperaments represent the middle of the oneirotonic spectrum (with the L/s ratio ranging from 3/2 to 3/1); [[13edo]] represents both temperaments, with a L/s ratio of 2/1. This is analogous to how in the diatonic spectrum, the [[19edo]]-to-[[17edo]]-range has the least extreme ratio of large to small step sizes, with [[12edo]] representing both [[meantone]] (19edo to 12edo) and [[pythagorean]]/[[neogothic]] (12edo to 17edo).


More extreme oneirotonic temperaments include:
=== Modes ===
* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.33¢ (17\45) and 457.14¢ (8\21). These have near-equal L/s ratios of 6/5 to 3/2.
{{MOS mode degrees}}
* [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5). While this is a harmonically accurate temperament, with 4 generators reaching [[3/2]] and -3 generators [[7/4]], it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.


In the past, 5L 3s has been viewed as a MOS of the low-accuracy 5-limit temperament [[father]]. This viewpoint is increasingly considered obsolete, but "father" is still sometimes used for both the 5L 3s oneirotonic and the 3L 2s oneiro-pentatonic.
==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}


== Scale tree ==
== Tunings==
{| class="wikitable" style="text-align:center;"
=== Simple tunings ===
|-
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.
! colspan="5" | generator
! | tetrachord
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| | 1 0 1
|  | 480.000
|  | 960.000
|  | 240.00
|  | 720.000
|  |
|-
| | 21\53
| |
| |
| |
| |
|  | 10 1 10
|  | 475.472
|  | 950.943
|  | 226.415
|  | 701.887
|  | Vulture/Buzzard is around here
|-
| | 19\48
| |
| |
| |
| |
|  | 9 1 9
|  | 475
|  | 950
|  | 225
|  | 700
|  |
|-
| | 17\43
| |
| |
| |
| |
|  | 8 1 8
|  | 474.419
|  | 948.837
|  | 223.256
|  | 697.674
|  |
|-
| | 15\38
| |
| |
| |
| |
|  | 7 1 7
|  | 473.684
|  | 947.368
|  | 221.053
|  | 694.737
|  |
|-
| | 13\33
| |
| |
| |
| |
|  | 6 1 6
|  | 472.727
|  | 945.455
|  | 218.181
|  | 690.909
|  |
|-
| | 11\28
| |
| |
| |
| |
|  | 5 1 5
|  | 471.429
|  | 942.857
|  | 214.286
|  | 685.714
|  |
|-
| | 9\23
| |
| |
| |
| |
|  | 4 1 4
|  | 469.565
|  | 939.130
|  | 208.696
|  | 678.261
|  | L/s = 4
|-
|
|16\41
|
|
|
|7 2 7
|468.293
|936.585
|204.878
|673.171
|Barbad is around here
|-
| | 7\18
| |
| |
| |
| |
|  | 3 1 3
|  | 466.667
|  | 933.333
|  | 200.000
|  | 666.667
|  | L/s = 3<br/>[[A-Team]] starts around here...
|-
| |
| | 19\49
| |
| |
| |
|  | 8 3 8
|  | 465.306
|  | 930.612
|  | 195.918
|  | 661.2245
| |
|-
| |
| |
| | 50\129
| |
| |
|  | 21 8 21
|  | 465.116
|  | 930.233
|  | 195.349
|  | 660.465
| |
|-
| |
| |
| |
| | 131\338
| |
|  | 55 21 55
|  | 465.089
|  | 930.1775
|  | 195.266
|  | 660.335
| |
|-
| |
| |
| |
| |
| | 212\547
|  | 89 34 89
|  | 465.082
|  | 930.1645
|  | 195.247
|  | 660.329
| |
|-
| |
| |
| |
| | 81\209
| |
|  | 34 13 34
|  | 465.072
|  | 930.1435
|  | 195.215
|  | 660.287
| |
|-
| |
| |
| | 31\80
| |
| |
|  | 13 5 13
|  | 465
|  | 930
|  | 195
|  | 660
| |
|-
| |
| | 12\31
| |
| |
| |
|  | 5 2 5
|  | 464.516
|  | 929.032
|  | 193.549
|  | 658.065
|  |
|-
| | 5\13
| |
| |
| |
| |
|  | 2 1 2
|  | 461.538
|  | 923.077
|  | 184.615
|  | 646.154
|  | ...and ends here<br/>Boundary of propriety (generators smaller than this are proper)<br/>[[5L_3s#Petrtri_.2813.2621.2C_2.5.9.11.13.17.29|Petrtri]] starts here...
|-
| |
| | 13\34
| |
| |
| |
|  | 5 3 5
|  | 458.824
|  | 917.647
|  | 176.471
|  | 635.294
|  |
|-
| |
| |
| | 34\89
| |
| |
|  | 13 8 13
|  | 458.427
|  | 916.854
|  | 175.281
|  | 633.708
|  |
|-
| |
| |
| |
| | 89\233
| |
|  | 34 21 34
|  | 458.369
|  | 916.738
|  | 175.107
|  | 633.473
|  |
|-
| |
| |
| |
| |
| | 233\610
|  | 89 55 89
|  | 458.361
|  | 916.721
|  | 175.082
|  | 633.443
|  | Golden oneirotonic; generator is 2 octaves minus logarithmic [[phi]]
|-
| |
| |
| |
| | 144\377
| |
|  | 55 34 55
|  | 458.355
|  | 916.711
|  | 175.066
|  | 633.422
|  |
|-
| |
| |
| | 55\144
| |
| |
|  | 21 13 21
|  | 458.333
|  | 916.666
|  | 175
|  | 633.333
|  |
|-
| |
| | 21\55
| |
| |
| |
|  | 8 5 8
|  | 458.182
|  | 916.364
|  | 174.545
|  | 632.727
|  |
|-
| | 8\21
| |
| |
| |
| |
|  | 3 2 3
|  | 457.143
|  | 914.286
|  | 171.429
|  | 628.571
|  | ...and ends here<br/> Optimum rank range (L/s=3/2) oneirotonic
|-
| | 11\29
| |
| |
| |
| |
|  | 4 3 4
|  | 455.172
|  | 910.345
|  | 165.517
|  | 620.690
|  | [[Tridec]] is around here
|-
| | 14\37
| |
| |
| |
| |
|  | 5 4 5
|  | 454.054
|  | 908.108
|  | 162.162
|  | 616.216
| |
|-
| | 17\45
| |
| |
| |
| |
|  | 6 5 6
|  | 453.333
|  | 906.667
|  | 160
|  | 613.333
| |
|-
| | 20\53
| |
| |
| |
| |
|  | 7 6 7
|  | 452.83
|  | 905.66
|  | 158.491
|  | 611.321
| |
|-
| | 23\61
| |
| |
| |
| |
|  | 8 7 8
|  | 452.459
|  | 904.918
|  | 157.377
|  | 609.836
| |
|-
| | 26\69
| |
| |
| |
| |
|  | 9 8 9
|  | 452.174
|  | 904.348
|  | 156.522
|  | 608.696
| |
|-
| | 29\77
| |
| |
| |
| |
|  | 10 9 10
|  | 451.948
|  | 903.896
|  | 155.844
|  | 607.792
| |
|-
| | 3\8
| |
| |
| |
| |
|  | 1 1 1
|  | 450.000
|  | 900.000
|  | 150.000
|  | 600.000
|  |
|}


==  Tuning ranges ==
{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
=== A-Team (13&18) ===
A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1.  


A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:  
=== Hypohard tunings ===
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Thus A-Team tempers out [[81/80]] like meantone does.
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:  
* The major mosthird (made of two large steps) is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.


EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].
* 13edo has characteristically small major mosseconds of about 185c. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all A-team tunings.
* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd a near-just 5/4.
* [[44edo]] (generator 17\44 = 463.64¢), [[57edo]] (generator 22\57 = 463.16¢), and [[70edo]] (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.


The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
{| class="wikitable right-2 right-3 right-4 right-5"
* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
|-
* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
!
* 31edo can be used to make the major 2-mosstep a near-just 5/4.
! [[13edo]]
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
! [[18edo]]
! [[31edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.9.5.21 subgroup)
|-
| generator (g)
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| 464.39
| 21/16
|-
| L (3g - octave)
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| 193.16
| 9/8, 10/9
|-
| s (-5g + 2 octaves)
| 1\13, 92.31
| 1\18, 66.67
| 2\31, 77.42
| 78.07
| 21/20
|}


Trivia: A-Team can be tuned by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic MOS. This produces a tuning close to 13edo.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}


=== Petrtri (13&21) ===
=== Hyposoft tunings ===
Petrtri tunings (with generator between 8\21 and 5\13) have less extreme L-to-s ratios than A-Team tunings, between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,  
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,  
* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92¢ to 114¢.
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢), and the temperament interprets it as both [[11/9]] and [[16/13]].
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).


The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri.
* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* [[13edo]] nominally supports it, but its approximation of 9:10:11:13 is quite weak and tempers 11/9 to a 369¢ submajor third, which may not be desirable.
* [[34edo]]'s 9:10:11:13 is even better.
* [[21edo]] is a much better petrtri tuning than 13edo, in terms of approximating 9:10:11:13. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).
* [[34edo]] is close to optimal for the temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.
* If you only care about optimizing 9:10:11:13, then [[55edo]]'s 21\55 (458.182¢) is even better, but 55 is a bit big for a usable edo.


The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[13edo]]
! [[21edo]]
! [[34edo]]
! Optimal (2.5.9.11.13.17 [[POTE]]) tuning
! JI intervals represented (2.5.9.11.13.17 subgroup)
|-
| generator (g)
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 459.15
| 13/10, 17/13, 22/17
|-
| L (3g - octave)
| 2\13, 184.62
| 3\21, 171.43
| 5\34, 176.47
| 177.45
| 10/9, 11/10
|-
| s (-5g + 2 octaves)
| 1\13, 92.31
| 2\21, 114.29
| 3\34, 105.88
| 104.25
| 18/17, 17/16
|}


=== Tridec (29&37) ===
{{MOS tunings
In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth, i.e. equates the oneirotonic large step to a [[porcupine]] generator. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-1 temperament|non-over-1 temperament]] that approximates the chord 5:7:11:13:15. Since it is the same as Petrtri when you only care about the 9:10:11:13 (R-M2-M3-M5), it can be regarded as a flatter variant of Petrtri (analogous to how septimal meantone and flattone are the same when you only consider how it maps 8:9:10:12).
| Step Ratios = Hyposoft
| JI Ratios =  
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}


The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
=== Parasoft and ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.


Based on the EDOs that support it, Tridec is essentially the same as 13-limit [[Ammonite]].
{{MOS tunings
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}


The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
=== Parahard tunings ===
{| class="wikitable right-2 right-3 right-4 right-5"
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).
|-
!
! [[21edo]]
! [[29edo]]
! [[37edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.3.7/5.11/5.13/5 subgroup)
|-
| generator (g)
| 8\21, 457.14
| 11\29, 455.17
| 14\37, 454.05
| 455.22
| 13/10
|-
| L (3g - octave)
| 3\21, 171.43
| 4\29, 165.52
| 5\37, 162.16
| 165.65
| 11/10
|-
| s (-5g + 2 octaves)
| 2\21, 114.29
| 3\29, 124.14
| 4\37, 129.73
| 123.91
| 14/13, 15/14
|}


=== Buzzard (48&53) ===
{{MOS tunings
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between A-Team and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into it's own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.
| JI Ratios =  
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}


Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
=== Ultrahard tunings ===
{{Main|5L&nbsp;3s/Temperaments#Buzzard}}


The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[38edo]]
! [[53edo]]
! [[63edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.3.5.7.13 subgroup)
|-
| generator (g)
| 15\38, 473.68
| 21\53, 475.47
| 25\63, 476.19
| 475.69
| 3/2 21/16
|-
| L (3g - octave)
| 7/38, 221.04
| 10/53, 226.41
| 12/63, 228.57
| 227.07
| 8/7
|-
| s (-5g + 2 octaves)
| 1/38 31.57
| 1/53 22.64
| 1/63 19.05
| 21.55
| 55/54 81/80 91/90
|}


== Notation==
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.  
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, with reference pitch J = 180 Hz, unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default.


Thus the [[13edo]] gamut is as follows:
Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.


'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M''' M&/N@ '''N/O@''' '''O/N&''' O&/P@ '''P''' '''Q''' Q&/J@ '''J'''
{{MOS tunings
| JI Ratios =
1/1;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}


The [[18edo]] gamut is notated as follows:
== Approaches ==
* [[5L&nbsp;3s/Temperaments]]


'''J''' Q&/K@ J&/L@ '''K''' '''L''' K&/M@ L& '''M''' N@ M&/O@ '''N''' '''O''' P@ O& '''P''' '''Q''' P&/J@ Q@ '''J'''
== Samples ==
 
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])
The [[21edo]] gamut:
 
'''J''' J& K@ '''K''' K&/L@ '''L''' L& M@ '''M''' M& N@ '''N''' N&/O@ '''O''' O& P@ '''P''' P&/Q@ '''Q''' Q& J@ '''J'''
 
Note: N is close to standard C, since the reference pitch 180 Hz for J was chosen to be nearly a pure 16/11 below standard 12edo C.
 
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! Octatonic interval category name
! Generators
! Notation of 2/1 inverse
! Octatonic interval category name
|-
| colspan="6" style="text-align:left" | The 8-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0
| J
| octave
|-
| 1
| M
| perfect oneirofourth (aka minor fourth, falling fourth)
| -1
| O
| perfect oneirosixth (aka major fifth, rising fifth)
|-
| 2
| P
| major oneiroseventh
| -2
| L
| minor oneirothird
|-
| 3
| K
| major oneirosecond
| -3
| Q
| minor oneiroeighth
|-
| 4
| N
| major oneirofifth (aka minor fifth, falling fifth)
| -4
| N@
| minor oneirofifth (aka major fourth, rising fourth)
|-
| 5
| Q&
| major oneiroeighth
| -5
| K@
| minor oneirosecond
|-
| 6
| L&
| major oneirothird
| -6
| P@
| minor oneiroseventh
|-
| 7
| O&
| augmented oneirosixth
| -7
| M@
| diminished oneirofourth
|-
| colspan="6" style="text-align:left" | The chromatic 13-note MOS also has the following intervals (from some root):
|-
| 8
| J&
| augmented unison
| -8
| J@
| diminished octave
|-
| 9
| M&
| augmented oneirofourth
| -9
| O@
| diminished oneirosixth
|-
| 10
| P&
| augmented oneiroseventh
| -10
| L@
| diminished oneirothird
|-
| 11
| K&
| augmented oneirosecond
| -11
| Q@
| diminished oneiroeighth
|-
| 12
| N&
| augmented oneirofifth
| -12
| N@@
| diminished oneirofifth
|}
 
== Key signatures ==
Flat keys:
* J@ Oneirominor, L@ Oneiromajor = N@, K@, P@, M@, J@, O@, L@, Q@
* M@ Oneirominor, O@ Oneiromajor = N@, K@, P@, M@, J@, O@, L@
* P@ Oneirominor, J@ Oneiromajor = N@, K@, P@, M@, J@, O@
* K@ Oneirominor, M@ Oneiromajor = N@, K@, P@, M@, J@
* N@ Oneirominor, P@ Oneiromajor = N@, K@, P@, M@
* Q Oneirominor, K@ Oneiromajor = N@, K@, P@
* L Oneirominor, N@ Oneiromajor = N@, K@
* O Oneirominor, Q Oneiromajor = N@
All-natural key signature:
* J Oneirominor, L Oneiromajor = no sharps or flats
Sharp keys:
* M Oneirominor, O Oneiromajor = Q&
* P Oneirominor, J Oneiromajor = Q&, L&
* K Oneirominor, M Oneiromajor = Q&, L&, O&
* N Oneirominor, P Oneiromajor = Q&, L&, O&, J&
* Q& Oneirominor, K Oneiromajor = Q&, L&, O&, J&, M&
** Enharmonic with J@ Oneirominor, L@ Oneiromajor in [[13edo]]
* L& Oneirominor, N Oneiromajor = Q&, L&, O&, J&, M&, P&
** Enharmonic with M@ Oneirominor, O@ Oneiromajor in 13edo
* O& Oneirominor, Q& Oneiromajor = Q&, L&, O&, J&, M&, P&, K&
** Enharmonic with P@ Oneirominor, J@ Oneiromajor in 13edo
* J& Oneirominor, L& Oneiromajor = Q&, L&, O&, J&, M&, P&, K&, N&
** Enharmonic with K@ Oneirominor, M@ Oneiromajor in 13edo
 
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands. (The names are by Cryptic Ruse.)
 
# Dylathian (də-LA(H)TH-iən): LLSLLSLS
# Illarnekian (ill-ar-NEK-iən): LLSLSLLS
# Celephaïsian (kel-ə-FAY-zhən): LSLLSLLS
# Ultharian (ul-THA(I)R-iən): LSLLSLSL
# Mnarian (mə-NA(I)R-iən): LSLSLLSL
# Kadathian (kə-DA(H)TH-iən): SLLSLLSL
# Hlanithian (lə-NITH-iən): SLLSLSLL
# Sarnathian (sar-NA(H)TH-iən): SLSLLSLL
 
The modes on the white keys JKLMNOPQJ are:
* J Ultharian
* K Hlanithian
* L Illarnekian
* M Mnarian
* N Sarnathian
* O Celephaïsian
* P Kadathian
* Q Dylathian
 
{| class="wikitable"
|-
|+ Table of modes (based on J, from brightest to darkest)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylathian
| J
| K
| L&
| M
| N
| O&
| P
| Q&
| (J)
|-
| Illarnekian
| J
| K
| L&
| M
| N
| O
| P
| Q&
| (J)
|-
| Celephaïsian
| J
| K
| L
| M
| N
| O
| P
| Q&
| (J)
|-
| Ultharian
| J
| K
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnarian
| J
| K
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Kadathian
| J
| K@
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Hlanithian
| J
| K@
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarnathian
| J
| K@
| L
| M@
| N@
| O
| P@
| Q
| (J)
|}
 
For classical-inspired functional harmony, we propose the terms ''(Functional) Oneiromajor'' and ''(Functional) Oneirominor'': Oneiromajor for Illarnekian where the 6th degree (the rising fifth) can be sharpened, and Oneirominor for Ultharian where the 8th degree (the leading tone) can be sharpened. The respective purposes of these alterations are:
# in Oneiromajor, to have both major (requiring a sharpened 6th degree) on the flat fourth "subdominant" and the sharp fifth as "dominant"
# in Oneirominor, to have both the flat 8th degree as the dominant of the "mediant" (relative major) and the sharp 8th degree as leading tone
 
In key signatures, Oneirominor should be treated as Ultharian and Oneiromajor should be treated as Illarnekian. Note that Oneiromajor and Oneirominor still have the relative major-minor relationship; they are related by a major mosthird, just like diatonic major/minor.
=== Alterations ===
==== Archeodim ====
We call the LSLLLSLS pattern (independently of modal rotation) '''archeodim''', because the "LLL" resembles the [[archeotonic]] scale in 13edo and the "LSLSLS" resembles the diminished scale. Archeodim is the most important oneirotonic [[MODMOS]] pattern (a MODMOS is a MOS with one or more alterations), because it allows one to evoke certain ana or kata diatonic modes where three whole steps in a row are important (Dorian, Phrygian, Lydian or Mixo) in an octatonic context. The MOS would not always be able to do this because it has at most two consecutive large steps. Archeodim modes exist in all oneirotonic tunings, since they use the same large and small steps as the oneirotonic scale itself.
 
As with the MOS, archeodim has four ana and four kata rotations:
* Ana:
** LLLSLSLS: Dylathian &4, Dylydian
** LLSLSLSL: Illarnekian @8, Illarmixian
** LSLLLSLS: Celephaïsian &6, Celdorian
** SLLLSLSL: Ultharian @2, Ulphrygian
* Kata:
** LSLSLLLS: Mnarian &8, Mnionian
** SLSLLLSL: Sarnathian &7, Sardorian
** LSLSLSLL: Mnarian @7, Mnaeolian
** SLSLSLLL: Sarnathian @6, Sarlocrian
 
{| class="wikitable"
|-
|+ Table of archeodim modes (based on J)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylydian
| J
| K
| L&
| M&
| N
| O&
| P
| Q&
| (J)
|-
| Illarmixian
| J
| K
| L&
| M
| N
| O
| P
| Q
| (J)
|-
| Celdorian
| J
| K
| L
| M
| N
| O&
| P
| Q&
| (J)
|-
| Ulphrygian
| J
| K@
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnionian
| J
| K
| L
| M
| N@
| O
| P
| Q&
| (J)
|-
| Sardorian
| J
| K@
| L
| M@
| N@
| O
| P
| Q
| (J)
|-
| Mnaeolian
| J
| K
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarlocrian
| J
| K@
| L
| M@
| N@
| O@
| P@
| Q
| (J)
|}
 
==== Other MODMOSes ====
Other potentially interesting oneirotonic MODMOSes (that do not use half-sharps or half-flats) are:
* the distorted harmonic minor LSLLSALS (A = aug 2nd = L + chroma)
* the distorted Freygish SASLSLLS
* Celephaïsian &4 &6 LsAsLsLs
 
== A-Team theory ==
Oneirotonic is often used as distorted diatonic. Because distorted diatonic modal harmony and functional harmony both benefit from a recognizable major third, the following theory essentially assumes an [[A-Team]] tuning, i.e. an oneirotonic tuning with generator between 5\13 and 7\18 (or possibly an approximation of such a tuning, such as a [[neji]]). The reader is encouraged to experiment and see what ideas work for other oneirotonic tunings.
=== Ana modes ===
We call modes with a major mos5th  ''ana modes'' (from Greek for 'up'), because the sharper 5th degree functions as a flattened melodic fifth when moving from the tonic up. The ana modes of the MOS are the 4 brightest modes, namely Dylathian, Illarnekian, Celephaïsian and Ultharian.
 
The ana modes have squashed versions of the classical major and minor pentachords R-M2-M3-P4-P5 and R-M2-m3-P4-P5 and can be viewed as providing a distorted version of classical diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context.
 
In pseudo-classical functional harmony, the 6th scale degree (either an augmented mossixth or a perfect mossixth) could be treated as mutable. The perfect mossixth would be used when invoking the diatonic V-to-I trope by modulating by a perfect mosfourth from the sixth degree "dominant". The augmented mossixth would be used when a major key needs to be used on the fourth degree "subdominant".
 
==== Pentatonic subsets ====
The ''Oneiro Falling Suspended Pentatonic'', i.e. P1-M2-P4-M5-M7 (on J, J-K-M-N-P), is also an important subset in ana modes: it roughly implies the "least" tonality (In particular, it only implies ana-ness, not major or minor tonality), and it sounds floaty, and suspended, much like suspended and quartal chords do in diatonic contexts. The ''Oneiro Rising Suspended Pentatonic'' P1-M2-P4-P6-M7 (J-K-M-O-P) can be used for similar effect.
 
Modes of the oneiro-pentatonic MOS:
# P1-M2-P4-M5-M7 Oneiro Falling Suspended Pentatonic
# P1-M2-P4-P6-M7 Oneiro Rising Suspended Pentatonic
# P1-m3-P4-P6-M7 Oneiro Symmetrical Pentatonic
# P1-m3-P4-P6-m8 Oneiro Expanding Quartal Pentatonic
# P1-m3-m5-P6-m8 Oneiro Diminished Pentatonic
 
==== Functional harmony ====
Oneiro has at least two different types of "V-to-I" resolution because of the two fifth sizes:
# One uses the sharp fifth as the "V" and uses a true major third. The sharp "V" voiceleads naturally to the flat fifth in the resolved falling tonic triad on the I: e.g. P6-M8-P2 > M5-P1-(M/m)3.
# One uses the flat fifth as the "V" and the chord on the "V" is a "false major triad" R-P4-P6 (root-falling 4th-rising 5th).
 
Some suggested basic ana functional harmony progressions are listed below, outlined very roughly. Note that VI, VII and VIII are sharp 5th, 6th-like and 7th-like degrees respectively. A Roman numeral without maj or min means either major or minor. The "Natural" Roman numerals follow the Illarnekian mode.
 
* I-IVmin-VImaj-I
* Imaj-VIImin-IVmin-Imaj
* Imin-@IIImaj-VImaj-Imaj
* Imin-@IIImaj-Vdim-VImaj-Imin
* Imin-@VIIImin-IIImaj-VImaj-Imin
* Imin-IVmin-@VIIImin-@IIImaj-VImaj-Imin
* Imin-IVmin-IIdim-VImaj-Imin
* Imin-IVmin-IIdim-@IIImaj-Imin
* I-VIImin-IImin-VImaj-I
* Imaj-VIImin-IVmin-VImaj-Imaj
* Modulations by major mos2nd:
** I-IV-VII-II
** I-IVmaj-II
** I-VIImin-II
* Modulations by major mos3rd:
** Modulate up major mos2nd twice
** Imin-VImin-III (only in 13edo)
** Imaj-&VImin-III (only in 13edo)
* Modulations by minor mos3rd:
** I-VI-@III
** I-IVmin-VImin-@VIIImaj-@III
 
Another Western-classical-influenced approach to oneirotonic chord progressions is to let the harmony emerge from counterpoint.
===== Samples =====
[[File:Oneiro Baroque Exercises 13edo.mp3‎]]
 
(A short contrapuntal 13edo keyboard exercise, meant to be played in all 13 keys. The first part is in Oneiromajor, i.e. Illarnekian with mutable 6th degree, and the second part is in Oneirominor, i.e. Celephaïsian with mutable 7th degree.)
 
[[File:Oneiro Baroque Exercises 18edo.mp3]]
 
([[18edo]] for comparison)
 
[[File:Oneiro Baroque Exercises 31edo.mp3]]
 
([[31edo]] for comparison)
 
[[File:Oneirotonic 3 part sample.mp3]]
 
‎(A rather classical-sounding 3-part harmonization of the ascending J Illarnekian scale; tuning is 13edo)
 
=== Kata modes ===
We call modes with a minor mos5th ''kata modes'' (from Greek for 'down'). The kata modes of the MOS are the 4 darkest modes, namely Mnarian, Kadathian, Hlanithian and Sarnathian. In kata modes, the melodically squashed fifth from the tonic downwards is the flatter 5th degree. Kata modes could be used to distort diatonic tropes that start from the tonic and work downwards or work upwards towards the tonic from below it. For example:
* Mnarian (LSLSLLSL) and Kadathian (SLLSLLSL) are kata-Mixolydians
* Hlanithian (SLLSLSLL) is a kata-melodic major (the 4th degree sounds like a major third; it's actually a perfect mosfourth.)
* Sarnathian (SLSLLSLL) is a kata-melodic minor (When starting from the octave above, the 4th degree sounds like a minor third; it's actually a diminished mosfourth.)
 
When used in an "ana" way, the kata modes are radically different in character than the ana modes. Particularly in 13edo and tunings close to it, the fifth and seventh scale degrees become the more concordant 11/8 and quasi-13/8 respectively, so they may sound more like stable scale functions. Hlanithian, in particular, may be like a more stable version of the Locrian mode in diatonic.
 
=== Chords ===
Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).
 
"Rising" means that a triad uses the perfect mos6th (major 5th); "falling" means that a triad uses a major mos5th (minor 5th)
 
* R-M3-M5: Falling Major Triad; Squashed Major Triad
* R-m3-M5: Falling Minor Triad; Squashed Minor Triad
* R-m3-m5: Squashed Dim Triad
* R-M3-A5: Squashed Aug Triad
* R-M3-M5-A6: Falling Major Triad Add6
* R-m3-M5-A6: Falling Minor Triad Add6
* R-M3-M5-M7: Falling Major Tetrad
* R-m3-M5-M7: Falling Minor Tetrad
* R-m3-m5-M7: Half-Diminished Tetrad
* R-m3-m5-m7: Orwell Tetrad, Diminished Tetrad
* R-M3-A6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
* R-M3-A6-M8: Sephiroth Triad Add7
* R-M3-A6-(M2)-(P4): Sephiroth Triad Add9 Sub11
* R-M3-A6-(m2)-(P4): Sephiroth Triad Addm9 Sub11
* R-M3-A6-(P4): Sephiroth Triad Sub11
* R-m3-P6: Rising Minor Triad; Squashed 1st Inversion Major Triad
* R-M3-P6: Rising Major Triad
* R-m3-M7: Minor add6 no5
* R-m3-m7: Minor addm6 no5
* R-m5-M7: Falling no3 add6
* R-m5-m7: Falling no3 add6
* R-M3-M8: Major 7th no5
* R-m3-M8: Minor Major 7th no5
* R-M3-M5-M8: Falling Major Seventh Tetrad
* R-m3-M5-M8: Falling Minor Major Seventh Tetrad
* R-M3-M7-M8: no5 Major Seventh Add6
* R-m3-M7-M8: no5 Minor Major Seventh Add6
* R-M3-P6-M8: Rising Major Seventh
* R-m3-P6-M8: Rising Oneiro Minor Major Seventh
* R-M3-(M2): Oneiro Major Add9
* R-m3-(M2): Oneiro Minor Add9
* R-M3-M5-(M2): Falling Major Triad Add9
* R-m3-M5-(M2): Falling Minor Triad Add9
* R-M3-(M2)-(P4): no5 Major Add9 Sub11
* R-m3-(M2)-(P4): no5 Minor Add9 Sub11
* R-m3-P6-M7-(M2)-(P4)-(A6)-(M8)
* R-M2-P4: Sus24 No5
* R-M2-M5: Falling Sus2 Triad
* R-P4-M5: Falling Sus4 Triad
* R-M2-P4-M5: Falling Sus24
* R-P4-M7: Oneiro Quartal Triad
* R-P4-M7-(M2): Oneiro Quartal Tetrad, Core Tetrad
* R-P4-M7-(M2)-(M5): Oneiro Quartal Pentad, Core Pentad
* R-P4-M7-(M2)-(M5)-(M8): Oneiro Quartal Hexad
* R-P4-M7-M8: Oneiro Quartal Seventh Tetrad
* R-P4-m8: Expanding Quartal Triad
* R-M2-P4-m8: Expanding Quartal Triad add2
* R-m3-P4-m8: Expanding Quartal Triad Addm3
* R-m5-m8: Contracting Quartal Triad
* R-m5-m7-m8: Contracting Quartal Triad Addm7
* R-M3-M5-m8: Falling Major Triad addm7
 
== Primodal theory ==
{{todo|Needs attention from primodalists}}
 
18edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).
 
31nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71¢) and 1\34 (35.29¢) are both dieses.
=== Primodal chords ===
Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by [[prime family]].
==== /11 ====
* 22:25:26:29:32:34:38:42:44 Undecimal Celephaïsian
* 22:25:26:29:32:34:38:40:44 Undecimal Ultharian
==== /13 ====
* 13:15:19 Tridecimal Falling Ultraminor Triad
* 13:16:19 Tridecimal Falling Submajor Triad
* 13:16:21 Tridecimal Squashed 1st Inversion Minor Triad
* 13:17:19 Tridecimal Naiadic Maj2; Tridecimal Falling Sus4
* 13:17:20 Tridecimal Rising Sus4
* 13:17:21 Tridecimal Squashed 2nd Inversion Major Triad
* 13:16:19:22 Tridecimal Falling Major Tetrad
* 26:29:38 Tridecimal Falling Sus2 Triad
* 26:31:38 Tridecimal Falling Bright Minor Triad
* 26:33:38 Tridecimal Falling Bright Major Triad
* 26:29:34:38 Tridecimal Falling Sus24
 
==== /17 ====
* 17:20:25 Septendecimal Falling Minor Triad
* 17:21:25 Septen Falling Major Triad
* 17:20:26 Septen Rising Minor Triad
* 17:20:25:29 Septen Falling Minor Tetrad
* 17:21:25:29 Septen Falling Major Tetrad
* 17:20:26:29 Septen Rising Minor Triad addM6
* 34:41:50 Septen Falling Bright Minor Triad
* 34:43:50 Septen Falling Octodecous Major Triad (''octodecous'' means '18edo-like')
* 34:40:47:55 Septen Orwell Tetrad
* 34:40:52:58:76:89:102:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-M14)
* 34:40:52:58:76:89:102:110:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-supmin13-M14)
* 34:40:50:58:89:102:129 (R-min3-f5-M6-M9-sub11-P12(rc)-M14)
* 34:40:50:58:89:102:110:129 (R-min3-f5-M6-M9-sub11-P12(rc)-supmin13-M14)
* 34:40:50:58:76:89:110:129 (R-m3-f5-M6-M9-sub11-supm13-M7)
* 34:40:50:58:76:89:102:110:129:208 (R-m3-f5-M6-M9-sub11-P12(rc)-supm13-M14-r19(rc^2))
* 34:38:40:44:49:52:58:64:68 Septen Celephaïsian
==== /19 ====
The notes 38:41:43:46:48:50:52:54:56:58:60:63:65:68:70:73:76 provide the best low complexity fit to oneirotonic (in particular, 18edo) in the [[prime family]] /19.
* 19:24:28 Novemdecimal Falling Bright Major Triad
* 19:23:28 Novem Falling Supraminor Triad
* 19:22:28 Novem Falling Ultraminor Triad
* 19:24:29 Novem Rising Major Triad
* 19:24:30 Novem Augmented Triad
* 19:24:43 Novem Major no5 add9
* 19:24:43:50 Novem Major no5 add9sub11
* 19:24:28:43:50 Novem Falling Major Triad add9 sub11
* 19:24:29:43:50 Novem Rising Major Triad add9 sub11
* 19:25:34 Novem Expanding Quartal
* 19:26:34 Novem Contracting Quartal
* 38:43:56 Novem Falling Minor Triad
* 38:45:56 Novemdecimal Falling Dark Major Triad
* 38:48:56:65 Novem Falling Major Tetrad
* 38:48:73 Novem Major Seventh no5
* 38:48:63 Novem Falling Major Triad
* 38:50:65 Novem Oneiro Quartal Triad
* 38:50:65:73 Novem Oneiro Quartal Seventh Tetrad
* 38:50:65:86 Novem Oneiro Core Tetrad
* 38:50:65:86:112 Novem Oneiro Core Pentad
* 38:50:65:86:112:146 Novem Oneiro Core Hexad
* 38:50:63 Novem Squashed First Inversion Neutral Triad
* 38:43:45:50:56:58:65:72:76 Novem Bright Celephaïsian
* 38:42:44:49:55:58:65:72:76 Novem Dark Celephaïsian
==== /23 ====
23(2:4) has many oneiro pitches, some close to 13edo, and some close to 18edo:
46:48:50:51:52:54:56:57:58:60:63:65:67:68:70:73:74:76:79:82:83:85:87:88:92
 
* 23:27:30 Vicesimotertial Falling Min4 no5
* 23:27:30:35:44 Vice Falling Min4 addM5,M7
* 23:27:37 Vice Orwell Tetrad no4
* 23:29:34 Vice Octodecous Falling Major Triad
* 46:54:68 Vice Octodecous Falling Minor Triad
* 46:54:60:67 Vice Falling Min4
* 46:54:63 Vice Squashed Dim
* 46:54:63:68 Vice Oneiro Half-diminished Tetrad
* 46:54:63:74 Vice Orwell Tetrad
* 46:54:67 Vice Tridecous Falling Minor Triad (''tridecous'' means '13edo-like')
* 46:57:67 Vice Tridecous Falling Major Triad
* 46:54:67:78 Vice Tridecous Falling Minor Tetrad
* 46:57:67:78 Vice Tridecous Falling Major Tetrad
* 46:54:60:67:78 Vice Falling Minor Tetrad Add Min4
* 46:60:67 Vice Falling Sus4
* 46:54:60:67 Vice Falling Min3 Sus4
* 46:52:58:60:68:76:79:89:92 Vice Bright Dylathian
* 46:51:57:60:67:75:78:88:92 Vice Dark Dylathian
* 46:52:58:60:68:71:79:89:92 Vice Bright Illarnekian
* 46:51:57:60:67:70:78:88:92 Vice Dark Illarnekian
* 46:52:54:60:68:71:79:89:92 Vice Bright Celephaïsian
* 46:51:54:60:67:70:78:88:92 Vice Dark Celephaïsian
* 46:52:54:60:68:71:79:83:92 Vice Bright Ultharian
* 46:51:54:60:67:70:78:82:92 Vice Dark Ultharian
* 46:52:54:60:64:71:79:83:92 Vice Bright Mnarian
* 46:51:54:60:63:70:78:82:92 Vice Dark Mnarian
* 46:49:54:60:64:71:79:83:92 Vice Bright Kadathian
* 46:48:54:60:63:70:78:82:92 Vice Dark Kadathian
* 46:49:54:60:64:71:75:83:92 Vice Bright Hlanithian
* 46:48:54:60:63:70:74:82:92 Vice Dark Hlanithian
* 46:49:54:58:64:71:75:83:92 Vice Bright Sarn
* 46:48:54:57:63:70:74:82:92 Vice Dark Sarn
 
==== /29 ====
* 29:34:38 Vicesimononal Falling Sus4
* 29:34:42 Vicenon Falling Minor Triad
* 29:36:42 Vicenon Falling Major Triad
* 29:34:40:47 Vicenon Orwell Tetrad
* 29:38:52 Vicenon Expanding Quartal Triad
* 29:40:52 Vicenon Contracting Quartal Triad
* 29:38:65:84:99 Vicenon Oneiro Core Pentad
* 29:38:65:84:99:110 Vicenon Oneiro Core Hexad
* 58:65:76:84:99:116 Vicenon Oneiro Falling Suspended Pentatonic
* 58:65:76:89:99:116 Vicenon Oneiro Rising Suspended Pentatonic
* 58:72:76:89:99:116 Vicenon Oneiro Symmetrical Pentatonic
* 58:72:76:89:104:116 Vicenon Oneiro Expanding Quartal Pentatonic
* 58:72:80:89:104:116 Vicenon Oneiro Diminished Pentatonic
* 58:65:72:80:84:94:99:110:116 Vicenon Dylydian
* 58:65:72:76:84:94:99:110:116 Vicenon Dylathian
* 58:65:72:76:84:89:99:110:116 Vicenon Illarnekian
* 58:65:72:76:84:89:99:104:116 Vicenon Illarmixian
* 58:65:68:76:84:94:99:110:116 Vicenon Celdorian
* 58:65:68:76:84:89:99:110:116 Vicenon Celephaïsian
* 58:65:68:76:84:89:99:104:116 Vicenon Ultharian
* 58:65:68:76:80:89:99:104:116 Vicenon Mnarian
* 58:65:68:76:80:89:99:110:116 Vicenon Mnionian
* 58:65:68:76:80:89:94:104:116 Vicenon Mnaeolian
* 58:61:68:76:80:89:99:104:116 Vicenon Kadathian
* 58:61:68:76:84:89:99:104:116 Vicenon Ulphrygian
* 58:61:68:76:80:89:94:104:116 Vicenon Hlanithian
* 58:61:68:72:80:89:99:104:116 Vicenon Sardorian
* 58:61:68:72:80:89:94:104:116 Vicenon Sarnathian
* 58:61:68:72:80:84:94:104:116 Vicenon Sarlocrian
==== /47 ====
* 47:52:55:61:68:72:80:89:94 Quadseptimal Celephaïsian
==== /61 ====
* 61:68:72:80:89:93:104:116:122 Sessantunesimal Celephaïsian
 
==== Over small prime multiples ====
 
=== Some oneirotonic nejis ===
The reader is encouraged to tweak these nejis and add more nejis that they like.
==== 13nejis ====
# '''58''':61:65:'''68''':72:'''76''':80:84:89:94:99:'''104''':110:116 - A low-complexity 13neji; has /13, /17, /19, and /29 prime modes
#* For lower complexity, can use 64 instead of 65 or 100 instead of 99
# 92:97:102:108:114:120:127:134:141:149:157:165:174:184 - Vice 13neji
 
==== 18nejis ====
# '''92''':96:100:'''104''':108:112:'''116''':120:125:130:'''136''':141:146:'''152''':158:164:170:177:184 - 18neji with /13, /17, /19, /23, and /29 prime modes
 
==== 21nejis ====
# 128:132:137:141:146:151:156:161:166:172:178:184:190:197:204:210:217:224:232:240:248:256
 
==== 31nejis ====
# 92:94:96:98:101:103:105:108:110:113:115:118:120:123:126:129:132:135:138:141:144:147:150:154:157:161:165:168:172:176:180:184
 
==== 34nejis ====
 
== Rank-2 temperaments ==
Oneirotonic temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team (13&18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13&21) and Tridec (21&29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R-M2-M3-M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of [[flattone]]:  it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.
 
Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only [[harmonic entropy]] minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.
=== Tridec ===
Subgroup: 2.3.7/5.11/5.13/5
 
Period: 1\1
 
Optimal ([[POTE]]) generator: 455.2178
 
EDO generators: [[21edo|8\21]], [[29edo|11\29]], [[37edo|14\37]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
[[Comma]] list: 196/195, 847/845, 1001/1000
 
[[Mapping]] (for 2, 3, 7/5, 11/5, 13/5): [{{val|1 5 2 0 1}}, {{val|0 -9 -4 3 1}}]
 
Mapping generators: ~2, ~13/10
 
{{Vals|legend=1| 21, 29, 37 }}
 
</div></div>
 
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Tridec interpretations:
 
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! Degree
! Size in 21edo
! Size in 29edo
! Size in 37edo
! Size in POTE tuning
! Note name on Q
! class="unsortable"| Approximate ratios
! #Gens up
|-
| 1
| 0\21, 0.00
| 0\29, 0.00
| 0\37, 0.00
| 0.00
| Q
| 1/1
| 0
|-
| 2
| 3\21, 171.43
| 4\29, 165.52
| 5\37, 163.16
| 165.65
| J
| 11/10, 10/9
| +3
|-
| 3
| 6\21, 342.86
| 8\29, 331.03
| 10\37, 324.32
| 331.31
| K
| 11/9, 6/5
| +6
|-
| 4
| 8\21, 457.14
| 11\29, 455.17
| 14\37, 454.05
| 455.17
| L
| 13/10, 9/7
| +1
|-
| 5
| 11\21, 628.57
| 15\29, 620.69
| 19\37, 616.22
| 620.87
| M
| 13/9, 10/7
| +4
|-
| 6
| 14\21, 800.00
| 19\29, 786.21
| 23\37, 778.38
| 786.52
| N
| 11/7
| +7
|-
| 7
| 16\21, 914.29
| 22\29, 910.34
| 28\37, 908.11
| 910.44
| O
| 22/13
| +2
|-
| 8
| 19\21, 1085.71
| 26\29, 1075.86
| 33\37, 1070.27
| 1076.09
| P
| 13/7, 28/15
| +5
|}
 
=== Petrtri ===
Subgroup: 2.5.9.11.13.17
 
Period: 1\1
 
Optimal ([[POTE]]) generator: 459.1502


EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 100/99, 144/143, 170/169, 221/220
[[Mapping]] (for 2, 5, 9, 11, 13, 17): [{{val|1 5 7 5 6 6}}, {{val|0 -7 -10 -4 -6 -5}}]
Mapping generators: ~2, ~13/10
{{Vals|legend=1| 13, 21, 34 }}
</div></div>
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 13edo
! Size in 21edo
! Size in 34edo
! Size in POTE tuning
! Note name on Q
! class="unsortable"| Approximate ratios
! #Gens up
|-
| 1
| 0\13, 0.00
| 0\21, 0.00
| 0\34, 0.00
| 0.00
| Q
| 1/1
| 0
|-
| 2
| 2\13, 184.62
| 3\21, 171.43
| 5\34, 176.47
| 177.45
| J
| 10/9, 11/10
| +3
|-
| 3
| 4\13, 369.23
| 6\21, 342.86
| 10\34, 352.94
| 354.90
| K
| 11/9, 16/13
| +6
|-
| 4
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 459.15
| L
| 13/10, 17/13, 22/17
| +1
|-
| 5
| 7\13, 646.15
| 11\21, 628.57
| 18\34, 635.294
| 636.60
| M
| 13/9, 16/11, 23/16 (esp. 21edo)
| +4
|-
| 6
| 9\13, 830.77
| 14\21, 800.00
| 23\34, 811.77
| 814.05
| N
| 8/5
| +7
|-
| 7
| 10\13, 923.08
| 16\21, 914.29
| 26\34, 917.65
| 918.30
| O
| 17/10
| +2
|-
| 8
| 12\13, 1107.69
| 19\21, 1085.71
| 31\34, 1094.12
| 1095.75
| P
| 17/9, 32/17, 15/8
| +5
|}
=== A-Team ===
Subgroup: 2.5.9.21
Period: 1\1
Optimal ([[POTE]]) generator: 464.3865
EDO generators: [[13edo|5\13]], [[18edo|7\18]], [[31edo|12\31]], [[44edo|17\44]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 81/80, 1029/1024
[[Mapping]] (for 2, 5, 9, 21): [{{val|1 0 2 4}}, {{val|0 6 3 1}}]
Mapping generators: ~2, ~21/16
{{Vals|legend=1| 13, 18, 31, 44 }}
</div></div>
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their A-Team interpretations:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! Degree
! Size in 13edo
! Size in 18edo
! Size in 31edo
! Note name on Q
! class="unsortable"| Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! #Gens up
|-
| 1
| 0\13, 0.00
| 0\18, 0.00
| 0\31, 0.00
| Q
| 1/1
| 0
|-
| 2
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| J
| 9/8, 10/9
| +3
|-
| 3
| 4\13, 369.23
| 6\18, 400.00
| 10\31, 387.10
| K
| 5/4
| +6
|-
| 4
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| L
| 21/16, ''13/10''
| +1
|-
| 5
| 7\13, 646.15
| 10\18, 666.66
| 17\31, 658.06
| M
| ''13/9'', ''16/11''
| +4
|-
| 6
| 9\13, 830.77
| 13\18, 866.66
| 22\31, 851.61
| N
| ''13/8'', ''18/11''
| +7
|-
| 7
| 10\13, 923.08
| 14\18, 933.33
| 24\31, 929.03
| O
| 12/7
| +2
|-
| 8
| 12\13, 1107.69
| 17\18, 1133.33
| 29\31, 1122.58
| P
|
| +5
|}
<references/>
=== Buzzard ===
Subgroup: 2.3.5.7
Period: 1\1
Optimal ([[POTE]]) generator: ~21/16 = 475.636
EDO generators: [[38edo|15\38]], [[43edo|17\43]], [[48edo|19\48]], [[53edo|21\53]], [[58edo|23\58]], [[63edo|25\63]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Commas: 1728/1715, 5120/5103
Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
Mapping generators: ~2, ~21/16
Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
{{Vals| 48, 53, 111, 164d, 275d}}
Badness: 0.0480
</div></div>
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 38edo
! Size in 53edo
! Size in 63edo
! Size in POTE tuning
! Note name on Q
! class="unsortable"| Approximate ratios
! #Gens up
|-
| 1
| 0\38, 0.00
| 0\53, 0.00
| 0\63, 0.00
| 0.00
| Q
| 1/1
| 0
|-
| 2
| 7\38, 221.05
| 10\53, 226.42
| 12\63, 228.57
| 227.07
| J
| 8/7
| +3
|-
| 3
| 14\38, 442.10
| 20\53, 452.83
| 24\63, 457.14
| 453.81
| K
| 13/10, 9/7
| +6
|-
| 4
| 15\38, 473.68
| 21\53, 475.47
| 25\63, 476.19
| 475.63
| L
| 21/16
| +1
|-
| 5
| 22\38, 694.73
| 31\53, 701.89
| 37\63, 704.76
| 702.54
| M
| 3/2
| +4
|-
| 6
| 29\38, 915.78
| 41\53, 928.30
| 49\63, 933.33
| 929.45
| N
| 12/7, 22/13
| +7
|-
| 7
| 30\38, 947.36
| 42\53, 950.94
| 50\63, 952.38
| 951.27
| O
| 26/15
| +2
|-
| 8
| 37\38, 1168.42
| 52\53, 1177.36
| 62\63, 1180.95
| 1178.18
| P
| 108/55, 160/81
| +5
|}
== Samples ==
[[File:13edo Prelude in J Oneirominor.mp3]]
[[File:13edo Prelude in J Oneirominor.mp3]]


Line 1,750: Line 188:
[[File:A Moment of Respite.mp3]]
[[File:A Moment of Respite.mp3]]


(13edo, L Illarnekian)
(13edo, L Ilarnekian)


[[File:Lunar Approach.mp3]]
[[File:Lunar Approach.mp3]]
Line 1,756: Line 194:
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
(by [[Igliashon Jones]], 13edo, J Celephaïsian)


== See also ==
=== 13edo Oneirotonic Modal Studies ===
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
 
== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}


[[Category:Scales]]
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Mos]]
[[Category:Pages with internal sound examples]]
[[Category:MOS scales]]
 
[[Category:Abstract MOS patterns]][[Category:Oneirotonic]]

Latest revision as of 13:59, 5 May 2025

↖ 4L 2s ↑ 5L 2s 6L 2s ↗
← 4L 3s 5L 3s 6L 3s →
↙ 4L 4s ↓ 5L 4s 6L 4s ↘ 
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Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 3\8 to 2\5 (450.0 ¢ to 480.0 ¢)
Dark 3\5 to 5\8 (720.0 ¢ to 750.0 ¢)
TAMNAMS information
Name oneirotonic
Prefix oneiro-
Abbrev. onei
Related MOS scales
Parent 3L 2s
Sister 3L 5s
Daughters 8L 5s, 5L 8s
Neutralized 2L 6s
2-Flought 13L 3s, 5L 11s
Equal tunings
Equalized (L:s = 1:1) 3\8 (450.0 ¢)
Supersoft (L:s = 4:3) 11\29 (455.2 ¢)
Soft (L:s = 3:2) 8\21 (457.1 ¢)
Semisoft (L:s = 5:3) 13\34 (458.8 ¢)
Basic (L:s = 2:1) 5\13 (461.5 ¢)
Semihard (L:s = 5:2) 12\31 (464.5 ¢)
Hard (L:s = 3:1) 7\18 (466.7 ¢)
Superhard (L:s = 4:1) 9\23 (469.6 ¢)
Collapsed (L:s = 1:0) 2\5 (480.0 ¢)
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).

5L 3s, named oneirotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 450 ¢ to 480 ¢, or from 720 ¢ to 750 ¢. 5L 3s can be seen as a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s).

Name

TAMNAMS suggests the temperament-agnostic name oneirotonic as the name of 5L 3s. The name was originally used as a name for the 5L 3s scale in 13edo. 'Oneiro' is sometimes used as a shortened form.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as father is technically an abstract regular temperament (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.

Scale properties

Intervals

Intervals of 5L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-oneirostep Perfect 0-oneirostep P0oneis 0 0.0 ¢
1-oneirostep Minor 1-oneirostep m1oneis s 0.0 ¢ to 150.0 ¢
Major 1-oneirostep M1oneis L 150.0 ¢ to 240.0 ¢
2-oneirostep Minor 2-oneirostep m2oneis L + s 240.0 ¢ to 300.0 ¢
Major 2-oneirostep M2oneis 2L 300.0 ¢ to 480.0 ¢
3-oneirostep Diminished 3-oneirostep d3oneis L + 2s 240.0 ¢ to 450.0 ¢
Perfect 3-oneirostep P3oneis 2L + s 450.0 ¢ to 480.0 ¢
4-oneirostep Minor 4-oneirostep m4oneis 2L + 2s 480.0 ¢ to 600.0 ¢
Major 4-oneirostep M4oneis 3L + s 600.0 ¢ to 720.0 ¢
5-oneirostep Perfect 5-oneirostep P5oneis 3L + 2s 720.0 ¢ to 750.0 ¢
Augmented 5-oneirostep A5oneis 4L + s 750.0 ¢ to 960.0 ¢
6-oneirostep Minor 6-oneirostep m6oneis 3L + 3s 720.0 ¢ to 900.0 ¢
Major 6-oneirostep M6oneis 4L + 2s 900.0 ¢ to 960.0 ¢
7-oneirostep Minor 7-oneirostep m7oneis 4L + 3s 960.0 ¢ to 1050.0 ¢
Major 7-oneirostep M7oneis 5L + 2s 1050.0 ¢ to 1200.0 ¢
8-oneirostep Perfect 8-oneirostep P8oneis 5L + 3s 1200.0 ¢

Generator chain

Generator chain of 5L 3s
Bright gens Scale degree Abbrev.
12 Augmented 4-oneirodegree A4oneid
11 Augmented 1-oneirodegree A1oneid
10 Augmented 6-oneirodegree A6oneid
9 Augmented 3-oneirodegree A3oneid
8 Augmented 0-oneirodegree A0oneid
7 Augmented 5-oneirodegree A5oneid
6 Major 2-oneirodegree M2oneid
5 Major 7-oneirodegree M7oneid
4 Major 4-oneirodegree M4oneid
3 Major 1-oneirodegree M1oneid
2 Major 6-oneirodegree M6oneid
1 Perfect 3-oneirodegree P3oneid
0 Perfect 0-oneirodegree
Perfect 8-oneirodegree		 P0oneid
P8oneid
−1 Perfect 5-oneirodegree P5oneid
−2 Minor 2-oneirodegree m2oneid
−3 Minor 7-oneirodegree m7oneid
−4 Minor 4-oneirodegree m4oneid
−5 Minor 1-oneirodegree m1oneid
−6 Minor 6-oneirodegree m6oneid
−7 Diminished 3-oneirodegree d3oneid
−8 Diminished 8-oneirodegree d8oneid
−9 Diminished 5-oneirodegree d5oneid
−10 Diminished 2-oneirodegree d2oneid
−11 Diminished 7-oneirodegree d7oneid
−12 Diminished 4-oneirodegree d4oneid

Modes

Scale degrees of the modes of 5L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (oneirodegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLsLLsLs Perf. Maj. Maj. Perf. Maj. Aug. Maj. Maj. Perf.
6|1 4 LLsLsLLs Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 7 LsLLsLLs Perf. Maj. Min. Perf. Maj. Perf. Maj. Maj. Perf.
4|3 2 LsLLsLsL Perf. Maj. Min. Perf. Maj. Perf. Maj. Min. Perf.
3|4 5 LsLsLLsL Perf. Maj. Min. Perf. Min. Perf. Maj. Min. Perf.
2|5 8 sLLsLLsL Perf. Min. Min. Perf. Min. Perf. Maj. Min. Perf.
1|6 3 sLLsLsLL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 6 sLsLLsLL Perf. Min. Min. Dim. Min. Perf. Min. Min. Perf.

Proposed mode names

The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.

Modes of 5L 3s
UDP Cyclic
order		 Step
pattern		 Mode names
7|0 1 LLsLLsLs Dylathian
6|1 4 LLsLsLLs Ilarnekian
5|2 7 LsLLsLLs Celephaïsian
4|3 2 LsLLsLsL Ultharian
3|4 5 LsLsLLsL Mnarian
2|5 8 sLLsLLsL Kadathian
1|6 3 sLLsLsLL Hlanithian
0|7 6 sLsLLsLL Sarnathian

Tunings

Simple tunings

The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


Simple Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Hard (3:1)
18edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\18 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 1\18 66.7 2\21 114.3
Major 1-oneirodegree M1oneid 2\13 184.6 3\18 200.0 3\21 171.4
Minor 2-oneirodegree m2oneid 3\13 276.9 4\18 266.7 5\21 285.7
Major 2-oneirodegree M2oneid 4\13 369.2 6\18 400.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 4\13 369.2 5\18 333.3 7\21 400.0
Perfect 3-oneirodegree P3oneid 5\13 461.5 7\18 466.7 8\21 457.1
Minor 4-oneirodegree m4oneid 6\13 553.8 8\18 533.3 10\21 571.4
Major 4-oneirodegree M4oneid 7\13 646.2 10\18 666.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 8\13 738.5 11\18 733.3 13\21 742.9
Augmented 5-oneirodegree A5oneid 9\13 830.8 13\18 866.7 14\21 800.0
Minor 6-oneirodegree m6oneid 9\13 830.8 12\18 800.0 15\21 857.1
Major 6-oneirodegree M6oneid 10\13 923.1 14\18 933.3 16\21 914.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 15\18 1000.0 18\21 1028.6
Major 7-oneirodegree M7oneid 12\13 1107.7 17\18 1133.3 19\21 1085.7
Perfect 8-oneirodegree P8oneid 13\13 1200.0 18\18 1200.0 21\21 1200.0

Hypohard tunings

Hypohard oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with meantone diatonic tunings:

  • The large step is a "meantone", around the range of 10/9 to 9/8.
  • The major 2-mosstep is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.

With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to 7/6.

EDOs that are in the hypohard range include 13edo, 18edo, and 31edo, and are associated with A-Team temperament.

  • 13edo has characteristically small 1-mossteps of about 185 ¢. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all hypohard tunings.
  • 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3 ¢, a perfect 5-mosstep) and falling fifths (666.7 ¢, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • 31edo can be used to make the major 2-mosstep a near-just 5/4.
  • 44edo (generator 17\44 = 463.64 ¢), 57edo (generator 22\57 = 463.16 ¢), and 70edo (generator 27\70 = 462.857 ¢}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.


Hypohard Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Semihard (5:2)
31edo 		Hard (3:1)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\31 0.0 0\18 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 2\31 77.4 1\18 66.7
Major 1-oneirodegree M1oneid 2\13 184.6 5\31 193.5 3\18 200.0
Minor 2-oneirodegree m2oneid 3\13 276.9 7\31 271.0 4\18 266.7
Major 2-oneirodegree M2oneid 4\13 369.2 10\31 387.1 6\18 400.0
Diminished 3-oneirodegree d3oneid 4\13 369.2 9\31 348.4 5\18 333.3
Perfect 3-oneirodegree P3oneid 5\13 461.5 12\31 464.5 7\18 466.7
Minor 4-oneirodegree m4oneid 6\13 553.8 14\31 541.9 8\18 533.3
Major 4-oneirodegree M4oneid 7\13 646.2 17\31 658.1 10\18 666.7
Perfect 5-oneirodegree P5oneid 8\13 738.5 19\31 735.5 11\18 733.3
Augmented 5-oneirodegree A5oneid 9\13 830.8 22\31 851.6 13\18 866.7
Minor 6-oneirodegree m6oneid 9\13 830.8 21\31 812.9 12\18 800.0
Major 6-oneirodegree M6oneid 10\13 923.1 24\31 929.0 14\18 933.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 26\31 1006.5 15\18 1000.0
Major 7-oneirodegree M7oneid 12\13 1107.7 29\31 1122.6 17\18 1133.3
Perfect 8-oneirodegree P8oneid 13\13 1200.0 31\31 1200.0 18\18 1200.0

Hyposoft tunings

Hyposoft oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,

  • The large step of oneirotonic tends to be intermediate in size between 10/9 and 11/10; the small step size is a semitone close to 17/16, about 92 ¢ to 114 ¢.
  • The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342 ¢) to 4\13 (369 ¢).
  • 21edo's P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71 ¢) and Baroque diatonic semitones (114.29 ¢, close to quarter-comma meantone's 117.11 ¢).
  • 34edo's 9:10:11:13 is even better.

This set of JI identifications is associated with petrtri temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)


Hyposoft Tunings of 5L 3s
Scale degree Abbrev. Soft (3:2)
21edo 		Semisoft (5:3)
34edo 		Basic (2:1)
13edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\21 0.0 0\34 0.0 0\13 0.0
Minor 1-oneirodegree m1oneid 2\21 114.3 3\34 105.9 1\13 92.3
Major 1-oneirodegree M1oneid 3\21 171.4 5\34 176.5 2\13 184.6
Minor 2-oneirodegree m2oneid 5\21 285.7 8\34 282.4 3\13 276.9
Major 2-oneirodegree M2oneid 6\21 342.9 10\34 352.9 4\13 369.2
Diminished 3-oneirodegree d3oneid 7\21 400.0 11\34 388.2 4\13 369.2
Perfect 3-oneirodegree P3oneid 8\21 457.1 13\34 458.8 5\13 461.5
Minor 4-oneirodegree m4oneid 10\21 571.4 16\34 564.7 6\13 553.8
Major 4-oneirodegree M4oneid 11\21 628.6 18\34 635.3 7\13 646.2
Perfect 5-oneirodegree P5oneid 13\21 742.9 21\34 741.2 8\13 738.5
Augmented 5-oneirodegree A5oneid 14\21 800.0 23\34 811.8 9\13 830.8
Minor 6-oneirodegree m6oneid 15\21 857.1 24\34 847.1 9\13 830.8
Major 6-oneirodegree M6oneid 16\21 914.3 26\34 917.6 10\13 923.1
Minor 7-oneirodegree m7oneid 18\21 1028.6 29\34 1023.5 11\13 1015.4
Major 7-oneirodegree M7oneid 19\21 1085.7 31\34 1094.1 12\13 1107.7
Perfect 8-oneirodegree P8oneid 21\21 1200.0 34\34 1200.0 13\13 1200.0

Parasoft and ultrasoft tunings

The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to porcupine temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a porcupine generator. The chord 10:11:13 is very well approximated in 29edo.


Soft Tunings of 5L 3s
Scale degree Abbrev. 6:5
45edo 		Supersoft (4:3)
29edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\45 0.0 0\29 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 5\45 133.3 3\29 124.1 2\21 114.3
Major 1-oneirodegree M1oneid 6\45 160.0 4\29 165.5 3\21 171.4
Minor 2-oneirodegree m2oneid 11\45 293.3 7\29 289.7 5\21 285.7
Major 2-oneirodegree M2oneid 12\45 320.0 8\29 331.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 16\45 426.7 10\29 413.8 7\21 400.0
Perfect 3-oneirodegree P3oneid 17\45 453.3 11\29 455.2 8\21 457.1
Minor 4-oneirodegree m4oneid 22\45 586.7 14\29 579.3 10\21 571.4
Major 4-oneirodegree M4oneid 23\45 613.3 15\29 620.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 28\45 746.7 18\29 744.8 13\21 742.9
Augmented 5-oneirodegree A5oneid 29\45 773.3 19\29 786.2 14\21 800.0
Minor 6-oneirodegree m6oneid 33\45 880.0 21\29 869.0 15\21 857.1
Major 6-oneirodegree M6oneid 34\45 906.7 22\29 910.3 16\21 914.3
Minor 7-oneirodegree m7oneid 39\45 1040.0 25\29 1034.5 18\21 1028.6
Major 7-oneirodegree M7oneid 40\45 1066.7 26\29 1075.9 19\21 1085.7
Perfect 8-oneirodegree P8oneid 45\45 1200.0 29\29 1200.0 21\21 1200.0

Parahard tunings

23edo oneiro combines the sound of neogothic tunings like 46edo and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as 46edo's neogothic major second, and is both a warped 22edo superpyth diatonic and a warped 24edo semaphore semiquartal (and both nearby scales are superhard MOSes).


Superhard Tuning of 5L 3s
Scale degree Abbrev. Superhard (4:1)
23edo
Steps ¢
Perfect 0-oneirodegree P0oneid 0\23 0.0
Minor 1-oneirodegree m1oneid 1\23 52.2
Major 1-oneirodegree M1oneid 4\23 208.7
Minor 2-oneirodegree m2oneid 5\23 260.9
Major 2-oneirodegree M2oneid 8\23 417.4
Diminished 3-oneirodegree d3oneid 6\23 313.0
Perfect 3-oneirodegree P3oneid 9\23 469.6
Minor 4-oneirodegree m4oneid 10\23 521.7
Major 4-oneirodegree M4oneid 13\23 678.3
Perfect 5-oneirodegree P5oneid 14\23 730.4
Augmented 5-oneirodegree A5oneid 17\23 887.0
Minor 6-oneirodegree m6oneid 15\23 782.6
Major 6-oneirodegree M6oneid 18\23 939.1
Minor 7-oneirodegree m7oneid 19\23 991.3
Major 7-oneirodegree M7oneid 22\23 1147.8
Perfect 8-oneirodegree P8oneid 23\23 1200.0

Ultrahard tunings

Main article: 5L 3s/Temperaments § Buzzard

Buzzard is a rank-2 temperament in the pseudocollapsed range. It represents the only harmonic entropy minimum of the oneirotonic spectrum.

In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. 23edo, 28edo and 33edo can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. 38edo & 43edo are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edo is where it really comes into its own in terms of harmonies, providing not only an excellent 3/2, but also 7/4 and archipelago harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.

Beyond that, it's a question of which intervals you want to favor. 53edo has an essentially perfect 3/2, 58edo gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while 63edo does the same for the basic 4:6:7 triad. You could in theory go up to 83edo if you want to favor the 7/4 above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.


Ultrahard Tunings of 5L 3s
Scale degree Abbrev. 7:1
38edo 		10:1
53edo 		12:1
63edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\38 0.0 0\53 0.0 0\63 0.0
Minor 1-oneirodegree m1oneid 1\38 31.6 1\53 22.6 1\63 19.0
Major 1-oneirodegree M1oneid 7\38 221.1 10\53 226.4 12\63 228.6
Minor 2-oneirodegree m2oneid 8\38 252.6 11\53 249.1 13\63 247.6
Major 2-oneirodegree M2oneid 14\38 442.1 20\53 452.8 24\63 457.1
Diminished 3-oneirodegree d3oneid 9\38 284.2 12\53 271.7 14\63 266.7
Perfect 3-oneirodegree P3oneid 15\38 473.7 21\53 475.5 25\63 476.2
Minor 4-oneirodegree m4oneid 16\38 505.3 22\53 498.1 26\63 495.2
Major 4-oneirodegree M4oneid 22\38 694.7 31\53 701.9 37\63 704.8
Perfect 5-oneirodegree P5oneid 23\38 726.3 32\53 724.5 38\63 723.8
Augmented 5-oneirodegree A5oneid 29\38 915.8 41\53 928.3 49\63 933.3
Minor 6-oneirodegree m6oneid 24\38 757.9 33\53 747.2 39\63 742.9
Major 6-oneirodegree M6oneid 30\38 947.4 42\53 950.9 50\63 952.4
Minor 7-oneirodegree m7oneid 31\38 978.9 43\53 973.6 51\63 971.4
Major 7-oneirodegree M7oneid 37\38 1168.4 52\53 1177.4 62\63 1181.0
Perfect 8-oneirodegree P8oneid 38\38 1200.0 53\53 1200.0 63\63 1200.0

Approaches

Samples

The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Ilarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

13edo Oneirotonic Modal Studies

Scale tree

Scale tree and tuning spectrum of 5L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 450.000 750.000 1:1 1.000 Equalized 5L 3s
17\45 453.333 746.667 6:5 1.200
14\37 454.054 745.946 5:4 1.250
25\66 454.545 745.455 9:7 1.286
11\29 455.172 744.828 4:3 1.333 Supersoft 5L 3s
30\79 455.696 744.304 11:8 1.375
19\50 456.000 744.000 7:5 1.400
27\71 456.338 743.662 10:7 1.429
8\21 457.143 742.857 3:2 1.500 Soft 5L 3s
29\76 457.895 742.105 11:7 1.571
21\55 458.182 741.818 8:5 1.600
34\89 458.427 741.573 13:8 1.625 Golden oneirotonic (458.3592 ¢)
13\34 458.824 741.176 5:3 1.667 Semisoft 5L 3s
31\81 459.259 740.741 12:7 1.714
18\47 459.574 740.426 7:4 1.750
23\60 460.000 740.000 9:5 1.800
5\13 461.538 738.462 2:1 2.000 Basic 5L 3s
Scales with tunings softer than this are proper
22\57 463.158 736.842 9:4 2.250
17\44 463.636 736.364 7:3 2.333
29\75 464.000 736.000 12:5 2.400
12\31 464.516 735.484 5:2 2.500 Semihard 5L 3s
31\80 465.000 735.000 13:5 2.600 Golden A-Team (465.0841 ¢)
19\49 465.306 734.694 8:3 2.667
26\67 465.672 734.328 11:4 2.750
7\18 466.667 733.333 3:1 3.000 Hard 5L 3s
23\59 467.797 732.203 10:3 3.333
16\41 468.293 731.707 7:2 3.500
25\64 468.750 731.250 11:3 3.667
9\23 469.565 730.435 4:1 4.000 Superhard 5L 3s
20\51 470.588 729.412 9:2 4.500
11\28 471.429 728.571 5:1 5.000
13\33 472.727 727.273 6:1 6.000
2\5 480.000 720.000 1:0 → ∞ Collapsed 5L 3s
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